Problem: Simplify; express your answer in exponential form. Assume $z\neq 0, q\neq 0$. $\dfrac{{(z^{-4}q^{4})^{3}}}{{(zq^{-3})^{-1}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(z^{-4}q^{4})^{3} = (z^{-4})^{3}(q^{4})^{3}}$ On the left, we have ${z^{-4}}$ to the exponent ${3}$ . Now ${-4 \times 3 = -12}$ , so ${(z^{-4})^{3} = z^{-12}}$ Apply the ideas above to simplify the equation. $\dfrac{{(z^{-4}q^{4})^{3}}}{{(zq^{-3})^{-1}}} = \dfrac{{z^{-12}q^{12}}}{{z^{-1}q^{3}}}$ Break up the equation by variable and simplify. $\dfrac{{z^{-12}q^{12}}}{{z^{-1}q^{3}}} = \dfrac{{z^{-12}}}{{z^{-1}}} \cdot \dfrac{{q^{12}}}{{q^{3}}} = z^{{-12} - {(-1)}} \cdot q^{{12} - {3}} = z^{-11}q^{9}$